Instability analysis and geometric perturbation theory to a mutual beneficial interaction between species with a higher order operator

José Luis Díaz Palencia; Abraham Otero; José Luis Díaz Palencia; Abraham Otero

doi:10.3934/math.2022947

AIMS Mathematics

2022, Volume 7, Issue 9: 17210-17224. doi: 10.3934/math.2022947

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Instability analysis and geometric perturbation theory to a mutual beneficial interaction between species with a higher order operator

José Luis Díaz Palencia ^{1,2
,
,},
Abraham Otero ¹

1.
Department of Information Technology, Escuela Politécnica Superior, Universidad San Pablo-CEU, CEU Universities, Campus Monteprincipe, Boadilla del Monte, Madrid, 28668, Spain
2.
Department of Mathematics and Education, Universidad a Distancia de Madrid, 28400, {Madrid}, Spain

Received: 28 March 2022 Revised: 24 June 2022 Accepted: 11 July 2022 Published: 22 July 2022
MSC : 35K92, 35K91, 35K55

The higher order diffusion can be understood as a generalization to the classical fickian diffusion. To account for such generalization, the Landau-Ginzburg free energy concept is applied leading to a fourth order spatial operator. This kind of diffusion induces a set of instabilities in the proximity of the critical points raising difficulties to study the convergence of Travelling Waves (TW) solutions. This paper aims at introducing a system of two species driven by a mutual interaction towards prospering and with a logistic term in their respective reactions. Previous to any analytical finding of TW solutions, the instabilities of such solutions are studied. Afterwards, the Geometric Perturbation Theory is applied to provide means to search for a linearized hyperbolic manifold in the proximity of the equilibrium points. The homotopy graphs for each of the flows to the hyperbolic manifolds are provided, so that analytical solutions can be obtained in the proximity of the critical points. Additionally, the set of eigenvalues in the homotopy graphs tend to cluster and synchronize for increasing values of the TW-speed.
- higher order diffusion,
- travelling waves,
- instabilities,
- propagation speed,
- geometric perturbation theory
Citation: José Luis Díaz Palencia, Abraham Otero. Instability analysis and geometric perturbation theory to a mutual beneficial interaction between species with a higher order operator[J]. AIMS Mathematics, 2022, 7(9): 17210-17224. doi: 10.3934/math.2022947

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Abstract

The higher order diffusion can be understood as a generalization to the classical fickian diffusion. To account for such generalization, the Landau-Ginzburg free energy concept is applied leading to a fourth order spatial operator. This kind of diffusion induces a set of instabilities in the proximity of the critical points raising difficulties to study the convergence of Travelling Waves (TW) solutions. This paper aims at introducing a system of two species driven by a mutual interaction towards prospering and with a logistic term in their respective reactions. Previous to any analytical finding of TW solutions, the instabilities of such solutions are studied. Afterwards, the Geometric Perturbation Theory is applied to provide means to search for a linearized hyperbolic manifold in the proximity of the equilibrium points. The homotopy graphs for each of the flows to the hyperbolic manifolds are provided, so that analytical solutions can be obtained in the proximity of the critical points. Additionally, the set of eigenvalues in the homotopy graphs tend to cluster and synchronize for increasing values of the TW-speed.

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